Question 8.6: Given the unity-feedback system of Figure 8.16, find the ang...

Using the poles and zeros of G(s) = (s + 2)/[(s + 3) (s²+ 2s + 2)] as plotted in Figure 8.17, we calculate the sum of angles drawn to a point ε close to the complex pole, −1 + j1, in the second quadrant. Thus,

− θ_{1}  −  θ_{2}  +  θ_{3}  −  θ_{4} =  − θ_{1}  −  90°  +  tan^{-1} (\frac{1}{1} )  −  tan^{-1} (\frac{1}{2} ) = 180°                     (8.46)

from which θ = − 251.6° = 108.4°. A sketch of the root locus is shown in Figure 8.17. Notice how the departure angle from the complex poles helps us to refine the shape.